Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is given by
in which is a Dirac delta function, , and a are some parameters, and x is the coordinate. You may find it useful to work in units h = 1 and m = 1/2 and to introduce k defined as E =—k2, where E < 0 is the energy of a bound state, so that k is real.
2.1 Find algebraic equations determining the energies (or k-values) of electron bound states for and arbitrary real 0 (positive or negative). How many bound states do you expect in this system? Describe the symmetry of their wave functions in terms of even and odd solutions.
2.2 Calculate the critical value a1 of the parameter a, at which an odd solution disappears.
2.3 Calculate the critical value a2 of the parameter a, at which an even solution disappears.
2.4 For β > 0, consider the limits of very large and very small a working out the analytic expressions for E (or k) for all bound states in this system, as well as their energy (or k) splittings at large a. Give a sketch of your result showing E (or k) versus a for all bound states in the system.